Problem 50 - Contemporary Precalculus

Question

In Exercises \(49-54\), prove the given identity. $$\sec (t+2 \pi)=\sec t[\text {Hint}: \text { See page } 500 .]$$

Step-by-Step Solution

Answer

Question: Prove the identity: \(\sec(t+2\pi) = \sec t\) Answer: Using the angle sum formula for the secant function and simplifying the expression, we showed that \(\sec(t+2\pi) = \sec t\).

Step 1: Recall the secant angle sum formula

Recall the formula for the sum of angles in the secant function: $$\sec(t + u) = \frac{\sec t \sec u}{1 - (\tan t)(\tan u)}$$ We will apply this formula to \(\sec(t+2\pi)\).

Step 2: Apply the angle sum formula to \(\sec(t+2\pi)\)

Now apply the sum of angles formula with \(t = t\) and \(u = 2 \pi\): $$\sec(t + 2\pi) = \frac{\sec(t) \sec(2\pi)}{1 - (\tan(t))(\tan(2\pi))}$$

Step 3: Simplify the expression

We now simplify the expression by evaluating \(\sec(2\pi)\) and \(\tan(2\pi)\): $$\sec(2\pi) = \sec(0) = 1$$ $$\tan(2\pi) = \tan(0) = 0$$ Substitute the values back into the equation: $$\sec(t + 2\pi) = \frac{\sec(t)(1)}{1 - (\tan(t))(0)}$$ Simplify the equation: $$\sec(t+2\pi) = \sec(t)$$ Hence, we have proven the identity and shown that \(\sec(t+2\pi) = \sec t\).

Key Concepts

Secant Function

Understanding the secant function is essential when diving into the world of trigonometry. Represented as \textbf{secant} or \( \text{sec}\), this function is one of the six fundamental trigonometric functions. Unlike its more familiar counterparts, sine and cosine, the secant function might not be as intuitive at first glance.

The secant function is actually the reciprocal of the cosine function. It is defined as \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \) for any angle \( x \), provided the cosine of \( x \) is not zero (since we cannot divide by zero). This function's values represent the hypotenuse's length divided by the adjacent side's length in a right-angled triangle.

In our exercise, we used the fact that \( \text{sec}(2\text{π}) = 1 \) since \( \text{cos}(2\text{π}) = 1 \), thereby simplifying our expression greatly. The fundamental period of secant, like cosine, is \( 2\text{π} \), meaning it repeats its pattern every \( 2\text{π} \) radians.

Angle Sum Formula

The angle sum formula for secant is a powerful tool that allows us to find the secant of the sum of two angles. It's based on the angle sum identities for sine and cosine but tailored for the secant function.

The general formula for the secant of the sum of two angles \( t \) and \( u \) is given by \( \text{sec}(t + u) = \frac{\text{sec}(t) \text{sec}(u)}{1 - (\tan(t))(\tan(u))} \). If you're wondering why this formula doesn't look like the common angle sum identities for sine or cosine, remember that it stems from the reciprocal nature of the secant function related to cosine. It's important to know this formula, as it enables us to break down more complex trigonometric calculations into simpler parts.

When we applied this formula in our exercise, setting \( u = 2\text{π} \), we could make use of the fact that the tangent of an angle with a whole multiple of \( 2\text{π} \) is zero, further simplifying our problem. The angle sum formula is a crucial identity for proving more advanced trigonometric properties.

Trigonometric Functions

Trigonometric functions form the cornerstone of trigonometry. They relate angles to the sides of a right-angled triangle but have applications far beyond triangles, such as oscillations and waves, engineering, and physics.

The primary trigonometric functions are sine (\( \text{sin} \)), cosine (\( \text{cos} \)), tangent (\( \text{tan} \)), along with their reciprocals, cosecant (\( \text{csc} = \frac{1}{\text{sin}} \)), secant (\( \text{sec} = \frac{1}{\text{cos}} \)), and cotangent (\( \text{cot} = \frac{1}{\text{tan}} \)). Each of these functions has its own unique graph, properties, and set of identities with which they interact.

In the context of the exercise, recognizing that secant and tangent functions are related to cosine and sine functions, respectively, and knowing their behavior at integral multiples of \( 2\text{π} \) was pivotal in solving the problem. Remember, a solid understanding of these basic trigonometric functions and their periodic properties is imperative for mastering more complex trigonometric equations and identities.